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Algorithmic randomness, quantum physics, and incompleteness
 PROCEEDINGS OF THE CONFERENCE “MACHINES, COMPUTATIONS AND UNIVERSALITY” (MCU’2004), LECTURES NOTES IN COMPUT. SCI. 3354
, 2004
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Most programs stop quickly or never halt
 Adv. Appl. Math
"... The aim of this paper is to provide a probabilistic, but nonquantum, analysis of the Halting Problem. Our approach is to have the probability space extend over both space and time and to consider the probability that a random Nbit program has halted by a random time. We postulate an a priori compu ..."
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Cited by 9 (3 self)
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The aim of this paper is to provide a probabilistic, but nonquantum, analysis of the Halting Problem. Our approach is to have the probability space extend over both space and time and to consider the probability that a random Nbit program has halted by a random time. We postulate an a priori computable probability distribution on all possible runtimes and we prove that given an integer k> 0, we can effectively compute a time bound T such that the probability that an Nbit program will eventually halt given that it has not halted by T is smaller than 2 −k. We also show that the set of halting programs (which is computably enumerable, but not computable) can be written as a disjoint union of a computable set and a set of effectively vanishing probability. Finally, we show that “long ” runtimes are effectively rare. More formally, the set of times at which an Nbit program can stop after the time 2 N+constant has effectively zero density. 1
References
, 2008
"... Chaitin’s “heuristic principle”, the theorems of a finitelyspecified theory cannot be significantly more complex than the theory itself was proved for an appropriate measure of complexity in [1]. The measure δ is a computable variation of the programsize complexity H: δ(x) = H(x) − x. The theo ..."
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Chaitin’s “heuristic principle”, the theorems of a finitelyspecified theory cannot be significantly more complex than the theory itself was proved for an appropriate measure of complexity in [1]. The measure δ is a computable variation of the programsize complexity H: δ(x) = H(x) − x. The theorems of a finitelyspecified, sound, consistent theory which is strong enough to include arithmetic have bounded δcomplexity, hence every sentence of the theory which is significantly more complex than the theory is unprovable. More precisely, according to Theorem 4.6 in [1], for any finitelyspecified, sound, consistent theory strong enough to formalize arithmetic (like ZermeloFraenkel set theory with choice or Peano Arithmetic) and for any Gödel numbering g of its wellformed formulae, we can compute a bound N such that no sentence x with complexity δg(x)> N can be proved in the theory; this phenomenon is independent on the choice of the Gödel numbering. Question 1. Find other natural measures of complexity for which Chaitin’s “heuristic principle ” holds true.
Most short programs halt quickly
, 2008
"... Since many realworld problems arising in the fields of compiler optimisation, automatised software engineering, formal proof systems, and so forth are equivalent to the Halting Problem—the most notorious undecidable problem—there is a growing interest, not only academically, in understanding the pr ..."
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Since many realworld problems arising in the fields of compiler optimisation, automatised software engineering, formal proof systems, and so forth are equivalent to the Halting Problem—the most notorious undecidable problem—there is a growing interest, not only academically, in understanding the problem better and in providing alternative solutions. Halting computations can be recognised by simply running them; the main difficulty is to detect nonhalting programs. For each program length on a given machine, there is an uncomputable “critical time ” after which no more programs of that length will halt. A quantum algorithm [7, 1] has been shown to solve the halting problem to any degree of certainty less than one and various experimental studies have proposed heuristics that apply to a majority of programs [4, 15]. Is it possible to have a classical effective way to describe this phenomenon? The aim of this paper is to provide a nonquantum analysis; our approach is to have the probability space extend over both space and time and to consider the probability that a random Nbit program has halted by